mws_gen_aae_spe_binaryrepresentation

mws_gen_aae_spe_binaryrepresentation - Chapter 01.04 Binary...

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01.04.1 Chapter 01.04 Binary Representation of Numbers After reading this chapter, you should be able to: 1. convert a base-10 real number to its binary representation, 2. convert a binary number to an equivalent base-10 number. In everyday life, we use a number system with a base of 10. For example, look at the number 257.56. Each digit in 257.56 has a value of 0 through 9 and has a place value. It can be written as 2 1 0 1 2 10 6 10 7 10 7 10 5 10 2 76 . 257 × + × + × + × + × = In a binary system, we have a similar system where the base is made of only two digits 0 and 1. So it is a base 2 system. A number like (1011.0011) in base-2 represents the decimal number as ( ) 1875 . 11 ) 2 1 2 1 2 0 2 0 ( ) 2 1 2 1 2 0 2 1 ( ) 0011 . 1011 ( 10 4 3 2 1 0 1 2 3 2 = × + × + × + × + × + × + × + × = in the decimal system. To understand the binary system, we need to be able to convert binary numbers to decimal numbers and vice-versa. We have already seen an example of how binary numbers are converted to decimal numbers. Let us see how we can convert a decimal number to a binary number. For example take the decimal number 11.1875. First, look at the integer part: 11. 1. Divide 11 by 2. This gives a quotient of 5 and a remainder of 1. Since the remainder is 1, 1 0 = a . 2. Divide the quotient 5 by 2. This gives a quotient of 2 and a remainder of 1. Since the remainder is 1, 1 1 = a . 3. Divide the quotient 2 by 2. This gives a quotient of 1 and a remainder of 0. Since the remainder is 0, 0 2 = a . 4. Divide the quotient 1 by 2. This gives a quotient of 0 and a remainder of 1. Since the remainder is , 1 3 = a . Since the quotient now is 0, the process is stopped. The above steps are summarized in Table 1.
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